Brainstorm how you would find the shaded area below.

Brainstorm how you would find the shaded area below.

Section 7.2: Areas in the Plane

Objective: Students will be able to…

• Find the area between two curves

Intro

We’ve already used integration to find the area under a curve bounded by the x-axis and the vertical lines defined by the limits of integration of the definite integral.

Area of f = b

a

dxxf )(

Area Between CurvesIf another curve g were to be drawn below f on the graph, the area between the two curves would be the integral (area) under curve f minus the integral (area) under curve g.

Area of f minus the area of g = dxxgxfb

a )()(

DefinitionIf f and g are continuous with f(x) ≥ g(x) throughout [a, b], then the area between the curves y = f(x) and y = g(x) from a to b is the integral of [f – g] from a to b,

A = dxxgxf

b

a )()(

ExampleFind the area of the region bounded by the graphs y=ex, y=x-2, x = -1, and x=2.

Write integral:

Evaluate:

ExampleFind the area of the region bounded by the graphs of y=x2 +2, y = -x, x = 0, and x = 1.

Area Enclosed by Intersecting Curves

When a region is enclosed by intersecting curves, the points of intersection give the limits of integration.

ExampleFind the area of the region enclosed by the parabolas y = x2 and y = 2x – x2.

Find where the 2 curves intersect:

Write integral:

Evaluate:

More about intersecting curves

If two curves intersect at more than two points, to find the area of the region between the curves, you must find all points of intersection and check to see which curve is above the other in each interval determined by these points.

Area = c

a

d

c

dxfgdxgf

ExampleFind the area of the region enclosed by

y=x4 – 2x2 and y = 2x2.

ExampleFind the area of the region between the graphs of

f(x) = 3x3 – x2 – 10x and g(x)=-x2+2x

Integrating with Respect to ySometimes the boundaries of a region are more easily

described as functions of y than by functions of x. To find the area of the region, integrate with respect to y.

A = 2

1

y

y

dycurveleftcurveright

ExampleFind the area of the region bounded by the graphs g(y)=3-y2 and f(y)=y+1.

(To graph, write explicitly)

Try these 3 examples.(Calculator)1. Find the area of the region enclosed by

y=3cosx and y=x2-1. (Use Nint to evaluate the integral)

2. Find the area of the region in the first quadrant that is bounded above by and below by the x –axis and the line y=x-2.

3. Find the area of the region bounded by graphs of f(y)=y(2-y) and g(y)= -y.

xy

Work for Example 1:

Work for Example 2:

Work for Example 3: